Solve for $x$ and $y$ using elimination. $\begin{align*}4x-3y &= 6 \\ -5x+y &= 1\end{align*}$
Answer: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $1$ and the bottom equation by $3$ $\begin{align*}4x-3y &= 6\\ -15x+3y &= 3\end{align*}$ Add the top and bottom equations. $-11x = 9$ Divide both sides by $-11$ and reduce as necessary. $x = -\dfrac{9}{11}$ Substitute $-\dfrac{9}{11}$ for $x$ in the top equation. $4( -\dfrac{9}{11})-3y = 6$ $-\dfrac{36}{11}-3y = 6$ $-3y = \dfrac{102}{11}$ $y = -\dfrac{34}{11}$ The solution is $\enspace x = -\dfrac{9}{11}, \enspace y = -\dfrac{34}{11}$.